Lecture Text Inductor, RCL, and
Transformer Circuits
Inductors, RCL Circuits and Transformers.
Simple RL filters work in similar but opposite manner to RC filters.
Replace the capacitor in a simple RC high pass filter with
an inductor and you get an RL low pass filter. Capacitors are less
expensive than inductors, and therefore RC circuits and not RL circuits are
generally used to create simple filters.
Complex pass band filters use both inductors and capacitors in conjunction
with resistance. Inductors and capacitors exhibit a property called resonance. Resistance is normally plotted on the x-axis, capacitance
reactance on the positive y-axis and inductive reactance on the negative
y-axis. Resistance and Capacitance reactance form two sides of triangle.
The total impedance for a resistor and capacitor in series is the vector sum
represented by the triangle hypotenuse. Resistance and inductive reactance form
two sides of triangle. The total impedance for a resistor and inductor in series
is the vector sum represented by the triangle hypotenuse. The capacitive
reactance phasor points in the opposite direction of the inductive reactance
phasor. Thus, the total reactance along the Y-axis can be determined algebraically.
The method I always used was simple. First I used my slide rule impedance
calculator to determine the capacitance and inductive reactance at the relevant
frequency. If the inductive reactance was greater than the capacitive
reactance, then the resultant reactance will be inductive. If the
capacitive reactance were greater than the inductive reactance, then the
resultant reactance will be capacitive. The magnitude of resultant
reactance is
determined by simple subtraction.
At some frequency the inductance reactance and
the capacitance reactance will be
equal, and their algebraic sum or vector sum for that matter will be zero.
That frequency is called the resonant frequency. There is no such thing as
a pure inductor coil, since the coil wire must have some resistance. Consider a small voltage applied to one end of
a capacitor. The capacitor is connected in
series with a coil, and the coils other end is tied to ground. As the frequency approaches
resonance the combined impedance of inductor and capacitor becomes very small, and the current increases
towards infinity. Thus, the voltage drop across the inductor and the
capacitor becomes very large. If you connected a high voltage scope probe to
the capacitor junction, you would measure a large voltage. Note that
the
high voltage scope probe implies a high resistance probe which will not effect
a circuit. It has been my experience that a times 100 scope probe will usually have an impedance of 100
meg-ohms, and a times 10 scope probe will usually have an impedance of ten
meg-ohms.
Let us consider the physics of the resonant circuit. When the voltage
reaches a peak, current goes to zero. Actually, current is at the point of
reversal, and all the energy of the oscillation is in the electrostatic field
between the capacitor plates. This electrostatic energy is potential energy. When the capacitor
discharges to zero, the current through the coil is at a maximum, and the energy
has changed from electrostatic potential energy to magnetic kinetic energy.
This is similar to the operation of a pendulum. At bottom of a pendulum's
arc, the speed and thus kinetic energy is at maximum, and the potential energy of
the system is
at a minimum. At the highest point of its arc the pendulum is stationary
for the instant that it reverses direction, and thus its kinetic energy is zero.
Resonance is important in communication, because it is the basis of tuned
circuits, oscillators and filters. A troubleshooter may be tracing a signal
expecting it to become large when it is amplified by a transistor or stepped up by
transformer. However, he might be confused by a small signal going to one end
of capacitor and coming out the other end many times larger in amplitude. You
may wonder where this large voltage came from and think that some wires must be
crossed. Nothing may be wrong at all, you may just be at the
junction of a capacitor and an inductor exhibiting resonance. I noticed,
while observing a complex filter that as the signal went from input to
output that it dipped lower than the final output and at the next stage it
jumped much higher than either the input or output. I was puzzled until I
realized that two resonant circuits with different resonant frequencies
were being cascaded to produce band pass filters. Formula for
resonance is derived below. Starting with the condition that
inductive reactance equals capacitive reactance at resonance, a little algebra
gives us the formula to calculate the resonant angular velocity of a LC series
circuit or an LC tank circuit.
A transformer is made starting with an
iron ring or other type ferromagnetic ring. A transformer is produced by
winding two or more coils around the ring. When a AC current is made to
flow through the input coil most of the magnetic flux is contained in the
ferromagnetic core. It follows that the same magnetic flux flows
through each coil wound around this common magnetic core. Each loop
of wire wound around the common core experiences the same rate of flux change as
the other loops, and thus the same induced E M F as the other loops.
Thus, the voltage output across each coil will be directly proportional to the
number of loops the coil has. No power is generated by a transformer; thus, when
the voltage to a circuit is stepped up, the current is stepped down. The
circuit below illustrates this rather well. Note that changing the
location of ground, changes the output of the transformer from one 240 volts
output to two 120 volt outputs. The 120 volt outputs are 180 degrees
out of phase.
My first eBook titled RCL Circuits
has many animations of filter and transformers in it. You can download it
for free. It is written in the Microsoft Help format, and it requires that you
keep two files in the same folder for it to work properly. Windows Vista
will not be delivered with support for the Help format. You can download
the Help Browser from Microsoft. Windows XP, 2000 and Windows 98 all come
with support for Help files embedded in the operating system.
